On the Structure of Semi-normal Operators
نویسنده
چکیده
1. Preliminaries. Only bounded operators on a Hilbert space § of elements x will be considered. If A is self-ad joint with the spectral resolution (1) A=f\dE(k), and if ^a = £>aC4) denotes the set of elements x for which ||E(X)#|| is an absolutely continuous function of X, then § 0 is a subspace; cf. [2, p. 240], [3, p. 436] and [6, p. 104]. If £ = £«, then A is called absolutely continuous. The one-dimensional Lebesgue measure of the spectrum of a self-adjoint operator A will be denoted by meas sp(A). An operator T on § is called semi-normal if (2) rr*~r*r=PèOorP^o. There will be proved the following result concerning such an operator. 2. Theorem. If T satisfies (2) and if 9ft = 90^ is the smallest subspace of § reducing T and containing the range of D, then (3) T+T* is absolutely continuous on 9W, andy if ffll denotes the orthogonal complement of '9JÏ (so that -ED? also reduces T), then (4) T is normal on $l. In addition y (5) 2i r | | l> | |^ | | r r* | | meassp ( r+r* ) , and the inequality (5) is optimal in the sense that there exist examples with D j^O for which (5) becomes an equality.
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تاریخ انتشار 2007